Assignment 4. Analysis exercises (Max 15 points)
1.The data
Explore the structure and the dimensions of the Boston data and describe the dataset briefly, assuming the reader has no previous knowledge of it. Details about the Boston dataset can be seen for example here. (0-1 points)
The Housing Values in Suburbs of Boston.
Dataset contains 14 colums including crime rate (mean 3.6, median 0.3), pupil-teacher ratio (mean 18.46, median 19.05) and non-retail business acres (mean 408, median 330) per town. Other interesting columns are for example distance from Boston center (mean 3.795, median 3.207), property tax-rate (mean 408.2, median 330) and amount of population of lower status (mean 12.65, median 11.36). There is no missingness in the data and rows 506 (towns?) in the data. All the data contain numerical variables, chas is binary.
Full list of columns are following (from here).
crim per capita crime rate by town.
zn proportion of residential land zoned for lots over 25,000 sq.ft.
indus proportion of non-retail business acres per town.
chas Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
nox nitrogen oxides concentration (parts per 10 million).
rm average number of rooms per dwelling.
age proportion of owner-occupied units built prior to 1940.
dis weighted mean of distances to five Boston employment centres.
rad index of accessibility to radial highways.
tax full-value property-tax rate per $10,000.
ptratio pupil-teacher ratio by town.
black 1000(Bk−0.63)21000(Bk−0.63)2 where BkBk is the proportion of blacks by town.
lstat lower status of the population (percent).
medv median value of owner-occupied homes in $1000s.
#install.packages("MASS")
library(MASS)
library(finalfit)
library(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:MASS':
##
## select## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(corrplot)## corrplot 0.92 loadeddata("Boston")
glimpse(Boston)## Rows: 506
## Columns: 14
## $ crim <dbl> 0.00632, 0.02731, 0.02729, 0.03237, 0.06905, 0.02985, 0.08829,…
## $ zn <dbl> 18.0, 0.0, 0.0, 0.0, 0.0, 0.0, 12.5, 12.5, 12.5, 12.5, 12.5, 1…
## $ indus <dbl> 2.31, 7.07, 7.07, 2.18, 2.18, 2.18, 7.87, 7.87, 7.87, 7.87, 7.…
## $ chas <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ nox <dbl> 0.538, 0.469, 0.469, 0.458, 0.458, 0.458, 0.524, 0.524, 0.524,…
## $ rm <dbl> 6.575, 6.421, 7.185, 6.998, 7.147, 6.430, 6.012, 6.172, 5.631,…
## $ age <dbl> 65.2, 78.9, 61.1, 45.8, 54.2, 58.7, 66.6, 96.1, 100.0, 85.9, 9…
## $ dis <dbl> 4.0900, 4.9671, 4.9671, 6.0622, 6.0622, 6.0622, 5.5605, 5.9505…
## $ rad <int> 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4,…
## $ tax <dbl> 296, 242, 242, 222, 222, 222, 311, 311, 311, 311, 311, 311, 31…
## $ ptratio <dbl> 15.3, 17.8, 17.8, 18.7, 18.7, 18.7, 15.2, 15.2, 15.2, 15.2, 15…
## $ black <dbl> 396.90, 396.90, 392.83, 394.63, 396.90, 394.12, 395.60, 396.90…
## $ lstat <dbl> 4.98, 9.14, 4.03, 2.94, 5.33, 5.21, 12.43, 19.15, 29.93, 17.10…
## $ medv <dbl> 24.0, 21.6, 34.7, 33.4, 36.2, 28.7, 22.9, 27.1, 16.5, 18.9, 15…ff_glimpse(Boston)## $Continuous
## label var_type n missing_n missing_percent mean sd min
## crim crim <dbl> 506 0 0.0 3.6 8.6 0.0
## zn zn <dbl> 506 0 0.0 11.4 23.3 0.0
## indus indus <dbl> 506 0 0.0 11.1 6.9 0.5
## chas chas <int> 506 0 0.0 0.1 0.3 0.0
## nox nox <dbl> 506 0 0.0 0.6 0.1 0.4
## rm rm <dbl> 506 0 0.0 6.3 0.7 3.6
## age age <dbl> 506 0 0.0 68.6 28.1 2.9
## dis dis <dbl> 506 0 0.0 3.8 2.1 1.1
## rad rad <int> 506 0 0.0 9.5 8.7 1.0
## tax tax <dbl> 506 0 0.0 408.2 168.5 187.0
## ptratio ptratio <dbl> 506 0 0.0 18.5 2.2 12.6
## black black <dbl> 506 0 0.0 356.7 91.3 0.3
## lstat lstat <dbl> 506 0 0.0 12.7 7.1 1.7
## medv medv <dbl> 506 0 0.0 22.5 9.2 5.0
## quartile_25 median quartile_75 max
## crim 0.1 0.3 3.7 89.0
## zn 0.0 0.0 12.5 100.0
## indus 5.2 9.7 18.1 27.7
## chas 0.0 0.0 0.0 1.0
## nox 0.4 0.5 0.6 0.9
## rm 5.9 6.2 6.6 8.8
## age 45.0 77.5 94.1 100.0
## dis 2.1 3.2 5.2 12.1
## rad 4.0 5.0 24.0 24.0
## tax 279.0 330.0 666.0 711.0
## ptratio 17.4 19.1 20.2 22.0
## black 375.4 391.4 396.2 396.9
## lstat 6.9 11.4 17.0 38.0
## medv 17.0 21.2 25.0 50.0
##
## $Categorical
## data frame with 0 columns and 506 rowssummary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.002. Graphical overview
Show a graphical overview of the data and show summaries of the variables in the data. Describe and interpret the outputs, commenting on the distributions of the variables and the relationships between them. (0-2 points)
Ggpairs. First I tried to the ggpairs plot matrix from previous weeks. The image is so bad that I had hard time to figuring out whats going on there. I added “proportions = auto” which helped little. Ggpairs show interesting correlations. For example variables listed there is positive correlation between crime rate and low status population ratio and negative correlation between crime rate and population of afro-americans.
Pairs. This plot gives same information than the last one. Ggpairs is bit more helpful when there is the correlation indicator.
Corrplot. This is gives same information than Ggpairs. Probably I will use corrplot in future than ggpairs because this is more tidy and it is easier to see which variables have higher correlation. For example dis (distanse from center) has negative correlation between indus (proportion of non-retail business acres), nox (itrogen oxides concentration) and age ( proportion of owner-occupied units built prior to 1940). Crime rate has positive correlations with rad (index of accessibility to radial highways) and tax (full-value property-tax rate per $10,000).
library(GGally) ## Loading required package: ggplot2## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(ggplot2)
# a plot matrix with ggpairs()
p2 <- ggpairs(Boston, mapping = aes(alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)), proportions = "auto")
p2
pairs(Boston)
# calculate the correlation matrix and round it
cor_matrix <- cor(Boston)
# print the correlation matrix
cor_matrix## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax ptratio
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431 0.2899456
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332 -0.3916785
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018 0.3832476
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652 -0.1215152
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320 0.1889327
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783 -0.3555015
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559 0.2615150
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158 -0.2324705
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819 0.4647412
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000 0.4608530
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304 1.0000000
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801 -0.1773833
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341 0.3740443
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593 -0.5077867
## black lstat medv
## crim -0.38506394 0.4556215 -0.3883046
## zn 0.17552032 -0.4129946 0.3604453
## indus -0.35697654 0.6037997 -0.4837252
## chas 0.04878848 -0.0539293 0.1752602
## nox -0.38005064 0.5908789 -0.4273208
## rm 0.12806864 -0.6138083 0.6953599
## age -0.27353398 0.6023385 -0.3769546
## dis 0.29151167 -0.4969958 0.2499287
## rad -0.44441282 0.4886763 -0.3816262
## tax -0.44180801 0.5439934 -0.4685359
## ptratio -0.17738330 0.3740443 -0.5077867
## black 1.00000000 -0.3660869 0.3334608
## lstat -0.36608690 1.0000000 -0.7376627
## medv 0.33346082 -0.7376627 1.0000000# visualize the correlation matrix
library(corrplot)
corrplot(cor_matrix, method="circle")
3. Standardize the dataset
Standardize the dataset and print out summaries of the scaled data. How did the variables change? Create a categorical variable of the crime rate in the Boston dataset (from the scaled crime rate). Use the quantiles as the break points in the categorical variable. Drop the old crime rate variable from the dataset. Divide the dataset to train and test sets, so that 80% of the data belongs to the train set. (0-2 points)
Standardize & scale the dataset. How did the variables change? First I notice that crime rate dropped from mean 3.61 and median 0.25651 to mean 0 and median -0.390280. Max values in every variable dropped significantly.
I tried corrplot out of curiosity and see no changes (no ****, sherlock).
# center and standardize variables
boston_scaled <- as.data.frame(scale(Boston))
boston_scaled$crim <- as.numeric(boston_scaled$crim)
# summaries of the scaled variables
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865glimpse(boston_scaled)## Rows: 506
## Columns: 14
## $ crim <dbl> -0.4193669, -0.4169267, -0.4169290, -0.4163384, -0.4120741, -0…
## $ zn <dbl> 0.28454827, -0.48724019, -0.48724019, -0.48724019, -0.48724019…
## $ indus <dbl> -1.2866362, -0.5927944, -0.5927944, -1.3055857, -1.3055857, -1…
## $ chas <dbl> -0.2723291, -0.2723291, -0.2723291, -0.2723291, -0.2723291, -0…
## $ nox <dbl> -0.1440749, -0.7395304, -0.7395304, -0.8344581, -0.8344581, -0…
## $ rm <dbl> 0.4132629, 0.1940824, 1.2814456, 1.0152978, 1.2273620, 0.20689…
## $ age <dbl> -0.11989477, 0.36680343, -0.26554897, -0.80908783, -0.51067434…
## $ dis <dbl> 0.1400749840, 0.5566090496, 0.5566090496, 1.0766711351, 1.0766…
## $ rad <dbl> -0.9818712, -0.8670245, -0.8670245, -0.7521778, -0.7521778, -0…
## $ tax <dbl> -0.6659492, -0.9863534, -0.9863534, -1.1050216, -1.1050216, -1…
## $ ptratio <dbl> -1.4575580, -0.3027945, -0.3027945, 0.1129203, 0.1129203, 0.11…
## $ black <dbl> 0.4406159, 0.4406159, 0.3960351, 0.4157514, 0.4406159, 0.41016…
## $ lstat <dbl> -1.07449897, -0.49195252, -1.20753241, -1.36017078, -1.0254866…
## $ medv <dbl> 0.15952779, -0.10142392, 1.32293748, 1.18158864, 1.48603229, 0…#cor_matrix2 <- cor(boston_scaled)
#cor_matrix2
#corrplot(cor_matrix2, method="circle")Create a categorical variable of the crime rate in the Boston dataset (from the scaled crime rate). Use the quantiles as the break points in the categorical variable.
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610# created a categorical variable 'crime': low, med_low, med_high and high
crime <- cut(boston_scaled$crim, breaks = bins, labels = c("low", "med_low", "med_high", "high"), include.lowest = TRUE)
# looking at the table of the new factor crime
crime## [1] low low low low low low med_low med_low
## [9] med_low med_low med_low med_low med_low med_high med_high med_high
## [17] med_high med_high med_high med_high med_high med_high med_high med_high
## [25] med_high med_high med_high med_high med_high med_high med_high med_high
## [33] med_high med_high med_high low med_low low med_low low
## [41] low med_low med_low med_low med_low med_low med_low med_low
## [49] med_low med_low med_low low low low low low
## [57] low low med_low med_low med_low med_low med_low med_low
## [65] low low low low med_low med_low med_low med_low
## [73] med_low med_low low med_low med_low med_low low med_low
## [81] low low low low low low low low
## [89] low low low low low low low med_low
## [97] med_low med_low low low med_low med_low med_low med_low
## [105] med_low med_low med_low med_low med_low med_high med_low med_low
## [113] med_low med_low med_low med_low med_low med_low med_low med_low
## [121] low low med_low med_low med_low med_low med_high med_high
## [129] med_high med_high med_high med_high med_high med_high med_high med_high
## [137] med_high med_high med_low med_high med_high med_high med_high high
## [145] med_high med_high med_high med_high med_high med_high med_high med_high
## [153] med_high med_high med_high med_high med_high med_high med_high med_high
## [161] med_high med_high med_high med_high med_high med_high med_high med_high
## [169] med_high med_high med_high med_high med_low med_low med_low low
## [177] low low low low low low med_low med_low
## [185] med_low low low low med_low med_low med_low low
## [193] med_low low low low low low low low
## [201] low low low low low med_low med_low med_low
## [209] med_low med_high med_low med_high med_low med_low med_high med_low
## [217] low low med_low med_low med_high med_high med_high med_high
## [225] med_high med_high med_high med_high med_high med_high med_high med_high
## [233] med_high med_high med_high med_high med_high med_high med_low med_low
## [241] med_low med_low med_low med_low med_low med_low med_high med_low
## [249] med_low med_low med_low med_low med_low med_high low low
## [257] low med_high med_high med_high med_high med_high med_high med_high
## [265] med_high med_high med_high med_high med_high med_low med_high med_low
## [273] med_low med_low low med_low med_low low low med_low
## [281] low low low low low low low low
## [289] low low low low low med_low low med_low
## [297] low med_low low low low low med_low med_low
## [305] low low low low med_high med_high med_high med_high
## [313] med_high med_high med_high med_low med_high med_low med_high med_high
## [321] med_low med_low med_high med_high med_high med_low med_high med_low
## [329] low low low low low low low low
## [337] low low low low low low low low
## [345] low low low low low low low low
## [353] low low low med_low high high high high
## [361] high high high high med_high high high high
## [369] high high high high high high high high
## [377] high high high high high high high high
## [385] high high high high high high high high
## [393] high high high high high high high high
## [401] high high high high high high high high
## [409] high high high high high high high high
## [417] high high high high high high high high
## [425] high high high high high high high high
## [433] high high high high high high high high
## [441] high high high high high high high high
## [449] high high high high high high high high
## [457] high high high high high high high high
## [465] high med_high high high high high high high
## [473] med_high high high high high high high high
## [481] high high high med_high med_high med_high high high
## [489] med_low med_low med_low med_low med_low med_low med_high med_low
## [497] med_high med_high med_low med_low med_low low low low
## [505] med_low low
## Levels: low med_low med_high highDrop the old crime rate variable from the dataset
# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
summary(boston_scaled)## zn indus chas nox
## Min. :-0.48724 Min. :-1.5563 Min. :-0.2723 Min. :-1.4644
## 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723 1st Qu.:-0.9121
## Median :-0.48724 Median :-0.2109 Median :-0.2723 Median :-0.1441
## Mean : 0.00000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723 3rd Qu.: 0.5981
## Max. : 3.80047 Max. : 2.4202 Max. : 3.6648 Max. : 2.7296
## rm age dis rad
## Min. :-3.8764 Min. :-2.3331 Min. :-1.2658 Min. :-0.9819
## 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049 1st Qu.:-0.6373
## Median :-0.1084 Median : 0.3171 Median :-0.2790 Median :-0.5225
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617 3rd Qu.: 1.6596
## Max. : 3.5515 Max. : 1.1164 Max. : 3.9566 Max. : 1.6596
## tax ptratio black lstat
## Min. :-1.3127 Min. :-2.7047 Min. :-3.9033 Min. :-1.5296
## 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049 1st Qu.:-0.7986
## Median :-0.4642 Median : 0.2746 Median : 0.3808 Median :-0.1811
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332 3rd Qu.: 0.6024
## Max. : 1.7964 Max. : 1.6372 Max. : 0.4406 Max. : 3.5453
## medv crime
## Min. :-1.9063 low :127
## 1st Qu.:-0.5989 med_low :126
## Median :-0.1449 med_high:126
## Mean : 0.0000 high :127
## 3rd Qu.: 0.2683
## Max. : 2.9865Divide the dataset to train and test sets, so that 80% of the data belongs to the train set.
# number of rows in the Boston dataset
n <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]4. Linear discriminant analysis
Fit the linear discriminant analysis on the train set. Use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables. Draw the LDA (bi)plot. (0-3 points)
# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2376238 0.2599010 0.2450495 0.2574257
##
## Group means:
## zn indus chas nox rm age
## low 0.9366381 -0.8979597 -0.10828322 -0.8650309 0.40032024 -0.8860226
## med_low -0.1205386 -0.2824813 0.02764047 -0.5364509 -0.08902241 -0.3250625
## med_high -0.3729012 0.1073640 0.16512651 0.3868752 0.10837110 0.3938960
## high -0.4872402 1.0170690 -0.04518867 1.1032617 -0.42935595 0.8253546
## dis rad tax ptratio black lstat
## low 0.8871997 -0.6899692 -0.7501294 -0.38699600 0.38994674 -0.75116352
## med_low 0.2854945 -0.5454537 -0.4803747 -0.09515703 0.31810027 -0.16350996
## med_high -0.3500123 -0.3995172 -0.3365550 -0.33872047 0.06266307 0.04119969
## high -0.8644505 1.6386213 1.5144083 0.78135074 -0.67162335 0.92669834
## medv
## low 0.47858202
## med_low 0.02863773
## med_high 0.18149342
## high -0.76206247
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.10277997 0.74999815 -0.91982374
## indus 0.03160664 -0.12906096 0.35354111
## chas -0.07520680 -0.02023296 0.17678097
## nox 0.34961890 -0.66683005 -1.56601435
## rm -0.09481904 -0.09268246 -0.11178694
## age 0.24432051 -0.30263354 -0.12835956
## dis -0.07507626 -0.14810129 -0.04073988
## rad 3.21861982 0.87006474 -0.13157442
## tax -0.05278143 -0.01411964 0.69886603
## ptratio 0.11176718 0.09034331 -0.35036288
## black -0.13193398 0.08129422 0.14715858
## lstat 0.23396754 -0.30027235 0.25891699
## medv 0.18242204 -0.47978877 -0.31461316
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9508 0.0362 0.0130# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
graphics::arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results (select both lines and execute them at the same time!)
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale = 1)
5. Predict & cross tabulate
Save the crime categories from the test set and then remove the categorical crime variable from the test dataset. Then predict the classes with the LDA model on the test data. Cross tabulate the results with the crime categories from the test set. Comment on the results. (0-3 points)
correct_classes <- test$crime
test <- dplyr::select(test, -crime)
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 20 10 1 0
## med_low 3 16 2 0
## med_high 0 9 17 1
## high 0 0 0 236. Standardize the dataset
Reload the Boston dataset and standardize the dataset (we did not do this in the Exercise Set, but you should scale the variables to get comparable distances).
data("Boston")
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# class of the boston_scaled object
class(boston_scaled)## [1] "matrix" "array"# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)Calculate the distances between the observations.
# euclidean distance matrix
dist_eu <- dist(Boston)
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.119 85.624 170.539 226.315 371.950 626.047# manhattan distance matrix
dist_man <- dist(Boston, method = "manhattan")
# look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 2.016 149.145 279.505 342.899 509.707 1198.265Run k-means algorithm on the dataset.
km <- kmeans(Boston, centers = 4)
# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)
# k-means clustering
km <- kmeans(Boston, centers = 4)
# plot the Boston dataset with clusters
pairs(Boston[6:10], col = km$cluster)
####
# k-means clustering
km <- kmeans(Boston, centers = 3)
# plot the Boston dataset with clusters
pairs(Boston[c("rm", "age", "dis", "crim")], col = km$cluster)
Investigate what is the optimal number of clusters and run the algorithm again.
set.seed(123)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
# k-means clustering
km <- kmeans(Boston, centers = 10)
# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)
The optimal number of clusters is when the total WCSS drops radically. In this example twcss drops when amount of clusters is two. I run the algorithm again with this.
k_max <- 2
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})Visualize the clusters (for example with the pairs() or ggpairs() functions, where the clusters are separated with colors) and interpret the results. (0-4 points)
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km <- kmeans(Boston, centers = 2)
# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)